Field
The invention relates to a microelectromechanical structure with a stator, and a rotor suspended for motion parallel to a first direction in relation to the stator.
Description of the Related Art
Micro-Electro-Mechanical Systems (MEMS) are miniaturized electro-mechanical systems that can be applied to quickly and accurately sense very small changes in physical properties. In many MEMS devices, sensing is based on detecting variations in capacitance.
In a parallel plate capacitor, capacitance is proportional to the area of overlap and inversely proportional to the separation between two capacitor plates. Parallel plate capacitors can be used to create closing gap structures, or area modulated structures.
FIG. 1A shows a configuration illustrating a parallel plate capacitor. In closing gap structures, capacitor plates move towards and away from each other in a direction denoted with x. Typically one of the plates is stationary, and the other plate moves closer to and further away from the other plate. The capacitance behavior can then be approximately modeled with equation (1)
                    C        =                  ∈                                    A                              d                -                x                                      +                          C              f                                                          (        1        )            where C is the capacitance, ε is permittivity, A a constant overlap area between the plates, d an initial gap between the plates, x a displacement from the initial gap position, and Cf a static stray capacitance.
FIG. 1B shows a configuration illustrating an area modulated structure, also known as a linear comb structure. In area modulated structures, the plates move parallel to each other and capacitance behavior can be modeled with equation (2)
                    C        =                  ∈                                                    h                ⁡                                  (                                      l                    +                    x                                    )                                            d                        +                          C              f                                                          (        2        )            where d is a constant gap between the plates, h a constant overlap dimension (height) of the plates, l an initial overlap length, x a displacement from the initial overlap length, and Cf a static stray capacitance.
The resonance frequency f of a harmonic oscillator is proportional to the electric spring constant. The relation can be written asf=1/(2π)√{square root over ( )}((km+ke)/m)  (6)where km is a mechanical spring constant, ke an electric spring constant, and m mass. By controlling the electric spring constant it is possible to tune the resonance frequency of the harmonic oscillator.
The potential energy E in a MEMS capacitor can be written asE=½kmx2−½CV2,  (3)whereinkm is the mechanical spring constant, x is a displacement from the initial capacitor structure, C is capacitance and V is voltage applied over the capacitor.
If the voltage over the capacitor is kept constant, the electric force Fe acting on the capacitor is obtained byFe=−∂E/∂x=½V2∂C/∂x.  (4)
An electric spring constant ke can then be obtained fromke=−∂Fe/∂x=−½V2∂C2/∂x2  (5)
The sign of the electric spring constant ke is thus dependent on the second derivative term ∂C2/∂x2. The curves in FIG. 2 illustrate variation of exemplary capacitances in the closing gap and linear comb structures of FIGS. 1A and 1B as a function of displacement from the initial position. The curves in FIG. 3 illustrate the corresponding behavior of the second derivative of these capacitances. It is seen that in commonly used prior art resonating structures, the second derivative is always positive (when parallel plate electrodes are used) or zero (in linear comb drive structures). However, for many applications further control to the electromechanical behavior of MEMS structures would be very valuable.